V is the subset of all polynomials ft a0 a1 t a2t2 in P2

V is the subset of all polynomials f(t) = a_0 + a_1 t + a_2t^2 in P_2 such that f\'(0) = 0. Show that V is a subspace of P_2.

Solution

f\'(t)=a1+2a2t

f\'(0)=a1=0

So, f(t)=a0+a2t^2

Hence, V =span{1,t^2}

HEnce, V is a subspace

We also prove it as follows

1. LEt, f=a0+a2t^2, g=b0+b2t^2 be in V

So, f+g=a0+b0+(a2+b2)t^2 is also in V

2. Let, f =a0+a2t^2 be in V and c be scalar

So, cf=ca0+ca2t^2 is also in V

Hence, V is subspace of P2